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Measures of Central Tendency. § 2.3. Population mean :. Sample mean :. “mu”. “ x - bar”. Mean.

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### § 2.4

### § 2.5

Sample mean:

“mu”

“x-bar”

MeanA measure of central tendency is a value that represents a typical, or central, entry of a data set. The three most commonly used measures of central tendency are the mean, the median, and the mode.

The mean of a data set is the sum of the data entries divided by the number of entries.

Mean

- Example:
- The following are the ages of all seven employees of a small company:

53 32 61 57 39 44 57

Calculatethe population mean.

Add the ages and divide by 7.

The mean age of the employees is 49 years.

Median

The median of a data set is the value that lies in the middle of the data when the data set is ordered. If the data set has an odd number of entries, the median is the middle data entry. If the data set has an even number of entries, the median is the mean of the two middle data entries.

- Example:
- Calculatethe median age of the seven employees.

53 32 61 57 39 44 57

To find the median, sort the data.

32 39 44 53 57 57 61

The median age of the employees is 53 years.

Mode

The mode of a data set is the data entry that occurs with the greatest frequency. If no entry is repeated, the data set has no mode. If two entries occur with the same greatest frequency, each entry is a mode and the data set is called bimodal.

Example:

Findthe mode of the ages of the seven employees.

53 32 61 57 39 44 57

The mode is 57 because it occurs the most times.

An outlier is a data entry that is far removed from the other entries in the data set.

Comparing the Mean, Median and Mode

- Example:
- A 29-year-old employee joins the company and the ages of the employees are now:

53 32 61 57 39 44 5729

Recalculate the mean, the median, and the mode. Which measure of central tendency was affected when this new age was added?

Mean = 46.5

The mean takes every value into account, but is affected by the outlier.

Median = 48.5

The median and mode are not influenced by extreme values.

Mode = 57

Weighted Mean

A weighted mean is the mean of a data set whose entries have varying weights. A weighted mean is given by

where w is the weight of each entry x.

Example:

Grades in a statistics class are weighted as follows:

Tests are worth 50% of the grade, homework is worth 30% of the grade and the final is worth 20% of the grade. A student receives a total of 80 points on tests, 100 points on homework, and 85 points on his final. What is his current grade?

Continued.

The mean of a frequency distribution for a sample is approximated by

where x and f are the midpoints and frequencies of the classes.

Mean of a Frequency DistributionExample:

The following frequency distribution represents the ages of 30 students in a statistics class. Find the mean of the frequency distribution.

Continued.

Shapes of Distributions

A frequency distribution is symmetric when a vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately the mirror images.

A frequency distribution is uniform (or rectangular) when all entries, or classes, in the distribution have equal frequencies. A uniform distribution is also symmetric.

A frequency distribution is skewed if the “tail” of the graph elongates more to one side than to the other. A distribution is skewedleft (negativelyskewed) if its tail extends to the left. A distribution is skewedright (positivelyskewed) if its tail extends to the right.

Skewed Left Distribution

5

Income

4

f

3

2

1

0

$25000

10 Annual Incomes

mean = $23,500

median = mode = $25,000

Mean < Median

Skewed Right Distribution

5

Income

4

f

3

2

1

0

$25000

10 Annual Incomes

mean = $121,500

median = mode = $25,000

Mean > Median

Summary of Shapes of Distributions

Uniform

Symmetric

Mean = Median

Skewed right

Skewed left

Mean > Median

Mean < Median

Range

The range of a data set is the difference between the maximum and minimum date entries in the set.

Range = (Maximum data entry) – (Minimum data entry)

Example:

The following data are the closing prices for a certain stock on ten successive Fridays. Find the range.

The range is 67 – 56 = 11.

x

56

58

61

63

67

DeviationThe deviation of an entry x in a population data set is the difference between the entry and the mean μof the data set.

Deviation of x = x – μ

Example:

The following data are the closing prices for a certain stock on five successive Fridays. Find the deviation of each price.

Deviation

x – μ

56 – 61 = – 5

58 – 61 = – 3

61 – 61 = 0

63 – 61 = 2

The mean stock price is

μ= 305/5 = 61.

67 – 61 = 6

Σx = 305

Σ(x – μ) = 0

“sigma”

Variance and Standard DeviationThe populationvariance of a population data set of N entries is

Population variance =

The populationstandard deviation of a population data set of N entries is the square root of the population variance.

Population standard deviation =

Finding the Population Standard Deviation

Guidelines

In Words In Symbols

- Find the mean of the population data set.
- Find the deviation of each entry.
- Square each deviation.
- Add to get the sum of squares.
- Divide by N to get the populationvariance.
- Find the square root of the variance to get the populationstandarddeviation.

Finding the Sample Standard Deviation

Guidelines

In Words In Symbols

- Find the mean of the sample data set.
- Find the deviation of each entry.
- Square each deviation.
- Add to get the sum of squares.
- Divide by n – 1 to get the samplevariance.
- Find the square root of the variance to get the samplestandarddeviation.

Finding the Population Standard Deviation

Example:

The following data are the closing prices for a certain stock on five successive Fridays. The population mean is 61. Find the population standard deviation.

SS2 = Σ(x – μ)2 = 74

σ $3.90

14

= 4

s = 1.18

= 4

s = 0

12

12

10

10

Frequency

Frequency

8

8

6

6

4

4

2

2

0

0

2

2

4

6

4

6

Data value

Data value

Interpreting Standard DeviationWhen interpreting standard deviation, remember that is a measure of the typical amount an entry deviates from the mean. The more the entries are spread out, the greater the standard deviation.

Empirical Rule (68-95-99.7%)

- Empirical Rule
- For data with a (symmetric) bell-shaped distribution, the standard deviation has the following characteristics.

- About 68% of the data lie within one standard deviation of the mean.
- About 95% of the data lie within two standard deviations of the mean.
- About 99.7% of the data lie within three standard deviation of the mean.

–3

–2

–1

0

1

2

3

4

2.35%

2.35%

Empirical Rule (68-95-99.7%)99.7% within 3 standard deviations

95% within 2 standard deviations

68% within 1 standard deviation

34%

34%

13.5%

13.5%

105

110

115

120

125

130

135

140

145

Using the Empirical Rule- Example:
- The mean value of homes on a street is $125 thousand with a standard deviation of $5 thousand. The data set has a bell shaped distribution. Estimate the percent of homes between $120 and $130 thousand.

μ – σ

μ

μ + σ

68% of the houses have a value between $120 and $130 thousand.

Chebychev’s Theorem

The Empirical Rule is only used for symmetric distributions.

Chebychev’s Theorem can be used for anydistribution, regardless of the shape.

For k = 2: In any data set, at least or 75%, of the data lie within 2 standard deviations of the mean.

For k = 3: In any data set, at least or 88.9%, of the data lie within 3 standard deviations of the mean.

Chebychev’s Theorem- The portion of any data set lying within k standard deviations (k > 1) of the mean is at least

45.8

48

50.2

52.4

54.6

56.8

59

Using Chebychev’s TheoremExample:

The mean time in a women’s 400-meter dash is 52.4 seconds with a standard deviation of 2.2 sec. At least 75% of the women’s times will fall between what two values?

At least 75% of the women’s 400-meter dash times will fall between 48 and 56.8 seconds.

Standard Deviation for Grouped Data

Sample standard deviation =

where n = Σf is the number of entries in the data set, and x is the data value or the midpoint of an interval.

Example:

The following frequency distribution represents the ages of 30 students in a statistics class. The mean age of the students is 30.3 years. Find the standard deviation of the frequency distribution.

Continued.

Standard Deviation for Grouped Data

The mean age of the students is 30.3 years.

The standard deviation of the ages is 10.2 years.

Quartiles

Median

Q1

Q2

Q3

0

25

50

75

100

Q3 is the median of the data above Q2.

Q1 is the median of the data below Q2.

The three quartiles, Q1, Q2, and Q3, approximately divide an ordered data set into four equal parts.

Finding Quartiles

Lower half

Upper half

Q2

Q3

Q1

Example:

The quiz scores for 15 students is listed below. Find the first, second and third quartiles of the scores.

28 43 48 51 43 30 55 44 48 33 45 37 37 42 38

Order the data.

28 30 3337 3738 42 43 43 44 45 48 48 51 55

About one fourth of the students scores 37 or less; about one half score 43 or less; and about three fourths score 48 or less.

Interquartile Range

The interquartile range (IQR) of a data set is the difference between the third and first quartiles.

Interquartile range (IQR) = Q3 – Q1.

Example:

The quartiles for 15 quiz scores are listed below. Find the interquartile range.

Q1 = 37

Q2 = 43

Q3 = 48

(IQR) = Q3 – Q1

The quiz scores in the middle portion of the data set vary by at most 11 points.

= 48 – 37

= 11

Box and Whisker Plot

A box-and-whisker plot is an exploratory data analysis tool that highlights the important features of a data set.

- The five-number summary is used to draw the graph.
- The minimum entry
- Q1
- Q2 (median)
- Q3
- The maximum entry

Example:

Use the data from the 15 quiz scores to draw a box-and-whisker plot.

28 30 3337 3738 42 4343 44 45 4848 51 55

Continued.

32

36

40

44

48

52

56

Box and Whisker Plot- Five-number summary
- The minimum entry
- Q1
- Q2 (median)
- Q3
- The maximum entry

28

37

43

48

55

Quiz Scores

28

37

43

48

55

Percentiles and Deciles

Fractiles are numbers that partition, or divide, an ordered data set.

Percentiles divide an ordered data set into 100 parts. There are 99 percentiles: P1, P2, P3…P99.

Deciles divide an ordered data set into 10 parts. There are 9 deciles: D1, D2, D3…D9.

A test score at the 80th percentile (P8), indicates that the test score is greater than 80% of all other test scores and less than or equal to 20% of the scores.

Standard Scores

The standardscore or z-score, represents the number of standard deviations that a data value, x, falls from the mean, μ.

Example:

The test scores for all statistics finals at Union College have a mean of 78 and standard deviation of 7. Find the z-score for

a.) a test score of 85,

b.) a test score of 70,

c.) a test score of 78.

Continued.

Standard Scores

Example continued:

a.) μ = 78, σ = 7, x = 85

This score is 1 standard deviation higher than the mean.

b.) μ = 78, σ = 7, x = 70

This score is 1.14 standard deviations lower than the mean.

c.) μ = 78, σ = 7, x = 78

This score is the same as the mean.

Relative Z-Scores

Example:

John received a 75 on a test whose class mean was 73.2 with a standard deviation of 4.5. Samantha received a 68.6 on a test whose class mean was 65 with a standard deviation of 3.9. Which student had the better test score?

John’s z-score

Samantha’s z-score

John’s score was 0.4 standard deviations higher than the mean, while Samantha’s score was 0.92 standard deviations higher than the mean. Samantha’s test score was better than John’s.

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